Long-term coexistence for a competitive system of spatially varying gradient reaction-diffusion equations

Spatial distribution of interacting chemical or biological species is Usually described by it system of reaction-diffusion equations. In this work we consider a system of two reaction-diffusion equations with spatially varying diffusion coefficients which are different for different species and with Forcing terms which are the gradient of a spatially varying potential. Such a system describes two competing biological species. We are interested in the possibility of long-term coexistence of the species in a bounded domain. Such long-term coexistence may be associated either with a periodic in time Solution (usually associated with a Hopf bifurcation, or with time-independent solutions. We prove that no periodic solution exists for the system. We also consider some steady states (the time-independent solutions) and examine their stability and bifurcations. (c) 2007 Elsevier Ltd. All rights reserved.